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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,2,Mod(26,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 36.4
Character \(\chi\) \(=\) 85.36
Dual form 85.2.l.a.26.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.213325 + 0.213325i) q^{2} +(0.980249 - 0.406032i) q^{3} -1.90899i q^{4} +(-0.382683 - 0.923880i) q^{5} +(0.295728 + 0.122495i) q^{6} +(-0.960473 + 2.31879i) q^{7} +(0.833883 - 0.833883i) q^{8} +(-1.32529 + 1.32529i) q^{9} +(0.115451 - 0.278722i) q^{10} +(2.25941 + 0.935880i) q^{11} +(-0.775110 - 1.87128i) q^{12} +5.61335i q^{13} +(-0.699548 + 0.289762i) q^{14} +(-0.750250 - 0.750250i) q^{15} -3.46219 q^{16} +(2.76113 - 3.06205i) q^{17} -0.565436 q^{18} +(-5.04243 - 5.04243i) q^{19} +(-1.76367 + 0.730537i) q^{20} +2.66297i q^{21} +(0.282343 + 0.681636i) q^{22} +(0.795280 + 0.329416i) q^{23} +(0.478830 - 1.15600i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(-1.19747 + 1.19747i) q^{26} +(-1.97910 + 4.77798i) q^{27} +(4.42653 + 1.83353i) q^{28} +(-1.43561 - 3.46587i) q^{29} -0.320094i q^{30} +(-2.07626 + 0.860015i) q^{31} +(-2.40634 - 2.40634i) q^{32} +2.59479 q^{33} +(1.24223 - 0.0641935i) q^{34} +2.50984 q^{35} +(2.52997 + 2.52997i) q^{36} +(4.71693 - 1.95382i) q^{37} -2.15135i q^{38} +(2.27920 + 5.50249i) q^{39} +(-1.08952 - 0.451294i) q^{40} +(4.72598 - 11.4095i) q^{41} +(-0.568078 + 0.568078i) q^{42} +(-1.85272 + 1.85272i) q^{43} +(1.78658 - 4.31319i) q^{44} +(1.73158 + 0.717244i) q^{45} +(0.0993804 + 0.239926i) q^{46} +2.30114i q^{47} +(-3.39381 + 1.40576i) q^{48} +(0.495478 + 0.495478i) q^{49} -0.301687 q^{50} +(1.46330 - 4.12268i) q^{51} +10.7158 q^{52} +(1.96204 + 1.96204i) q^{53} +(-1.44145 + 0.597069i) q^{54} -2.44557i q^{55} +(1.13268 + 2.73452i) q^{56} +(-6.99022 - 2.89544i) q^{57} +(0.433105 - 1.04561i) q^{58} +(-5.26206 + 5.26206i) q^{59} +(-1.43222 + 1.43222i) q^{60} +(-0.346822 + 0.837303i) q^{61} +(-0.626380 - 0.259455i) q^{62} +(-1.80017 - 4.34599i) q^{63} +5.89773i q^{64} +(5.18606 - 2.14814i) q^{65} +(0.553532 + 0.553532i) q^{66} -6.69889 q^{67} +(-5.84541 - 5.27096i) q^{68} +0.913326 q^{69} +(0.535411 + 0.535411i) q^{70} +(0.222439 - 0.0921372i) q^{71} +2.21028i q^{72} +(-2.47116 - 5.96591i) q^{73} +(1.42304 + 0.589441i) q^{74} +(-0.406032 + 0.980249i) q^{75} +(-9.62592 + 9.62592i) q^{76} +(-4.34022 + 4.34022i) q^{77} +(-0.687606 + 1.66003i) q^{78} +(13.5899 + 5.62912i) q^{79} +(1.32492 + 3.19865i) q^{80} -0.135560i q^{81} +(3.44210 - 1.42577i) q^{82} +(-9.82767 - 9.82767i) q^{83} +5.08358 q^{84} +(-3.88561 - 1.37916i) q^{85} -0.790460 q^{86} +(-2.81451 - 2.81451i) q^{87} +(2.66450 - 1.10367i) q^{88} -0.395163i q^{89} +(0.216383 + 0.522395i) q^{90} +(-13.0162 - 5.39148i) q^{91} +(0.628850 - 1.51818i) q^{92} +(-1.68606 + 1.68606i) q^{93} +(-0.490889 + 0.490889i) q^{94} +(-2.72894 + 6.58825i) q^{95} +(-3.33586 - 1.38176i) q^{96} +(3.42855 + 8.27725i) q^{97} +0.211396i q^{98} +(-4.23471 + 1.75407i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 24 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 32 q^{22} - 16 q^{23} - 8 q^{24} + 16 q^{26} + 24 q^{27} + 48 q^{28} - 8 q^{29} + 16 q^{34} - 32 q^{35}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213325 + 0.213325i 0.150843 + 0.150843i 0.778495 − 0.627651i \(-0.215984\pi\)
−0.627651 + 0.778495i \(0.715984\pi\)
\(3\) 0.980249 − 0.406032i 0.565947 − 0.234423i −0.0813177 − 0.996688i \(-0.525913\pi\)
0.647265 + 0.762265i \(0.275913\pi\)
\(4\) − 1.90899i − 0.954493i
\(5\) −0.382683 − 0.923880i −0.171141 − 0.413171i
\(6\) 0.295728 + 0.122495i 0.120731 + 0.0500082i
\(7\) −0.960473 + 2.31879i −0.363025 + 0.876420i 0.631830 + 0.775107i \(0.282304\pi\)
−0.994855 + 0.101312i \(0.967696\pi\)
\(8\) 0.833883 − 0.833883i 0.294822 − 0.294822i
\(9\) −1.32529 + 1.32529i −0.441765 + 0.441765i
\(10\) 0.115451 − 0.278722i 0.0365087 − 0.0881397i
\(11\) 2.25941 + 0.935880i 0.681239 + 0.282179i 0.696345 − 0.717707i \(-0.254808\pi\)
−0.0151056 + 0.999886i \(0.504808\pi\)
\(12\) −0.775110 − 1.87128i −0.223755 − 0.540192i
\(13\) 5.61335i 1.55686i 0.627729 + 0.778432i \(0.283985\pi\)
−0.627729 + 0.778432i \(0.716015\pi\)
\(14\) −0.699548 + 0.289762i −0.186962 + 0.0774422i
\(15\) −0.750250 − 0.750250i −0.193714 − 0.193714i
\(16\) −3.46219 −0.865549
\(17\) 2.76113 − 3.06205i 0.669673 − 0.742656i
\(18\) −0.565436 −0.133275
\(19\) −5.04243 − 5.04243i −1.15681 − 1.15681i −0.985157 − 0.171655i \(-0.945089\pi\)
−0.171655 − 0.985157i \(-0.554911\pi\)
\(20\) −1.76367 + 0.730537i −0.394369 + 0.163353i
\(21\) 2.66297i 0.581108i
\(22\) 0.282343 + 0.681636i 0.0601957 + 0.145325i
\(23\) 0.795280 + 0.329416i 0.165827 + 0.0686880i 0.464053 − 0.885807i \(-0.346395\pi\)
−0.298226 + 0.954495i \(0.596395\pi\)
\(24\) 0.478830 − 1.15600i 0.0977407 − 0.235967i
\(25\) −0.707107 + 0.707107i −0.141421 + 0.141421i
\(26\) −1.19747 + 1.19747i −0.234843 + 0.234843i
\(27\) −1.97910 + 4.77798i −0.380879 + 0.919522i
\(28\) 4.42653 + 1.83353i 0.836536 + 0.346505i
\(29\) −1.43561 − 3.46587i −0.266586 − 0.643597i 0.732732 − 0.680518i \(-0.238245\pi\)
−0.999318 + 0.0369210i \(0.988245\pi\)
\(30\) − 0.320094i − 0.0584409i
\(31\) −2.07626 + 0.860015i −0.372907 + 0.154463i −0.561263 − 0.827638i \(-0.689684\pi\)
0.188356 + 0.982101i \(0.439684\pi\)
\(32\) −2.40634 − 2.40634i −0.425385 − 0.425385i
\(33\) 2.59479 0.451694
\(34\) 1.24223 − 0.0641935i 0.213041 − 0.0110091i
\(35\) 2.50984 0.424240
\(36\) 2.52997 + 2.52997i 0.421661 + 0.421661i
\(37\) 4.71693 − 1.95382i 0.775459 − 0.321206i 0.0403776 − 0.999184i \(-0.487144\pi\)
0.735081 + 0.677979i \(0.237144\pi\)
\(38\) − 2.15135i − 0.348995i
\(39\) 2.27920 + 5.50249i 0.364965 + 0.881103i
\(40\) −1.08952 − 0.451294i −0.172268 − 0.0713559i
\(41\) 4.72598 − 11.4095i 0.738074 − 1.78187i 0.124518 − 0.992217i \(-0.460262\pi\)
0.613556 − 0.789651i \(-0.289738\pi\)
\(42\) −0.568078 + 0.568078i −0.0876564 + 0.0876564i
\(43\) −1.85272 + 1.85272i −0.282536 + 0.282536i −0.834120 − 0.551583i \(-0.814024\pi\)
0.551583 + 0.834120i \(0.314024\pi\)
\(44\) 1.78658 − 4.31319i 0.269337 − 0.650238i
\(45\) 1.73158 + 0.717244i 0.258129 + 0.106920i
\(46\) 0.0993804 + 0.239926i 0.0146528 + 0.0353751i
\(47\) 2.30114i 0.335655i 0.985816 + 0.167828i \(0.0536752\pi\)
−0.985816 + 0.167828i \(0.946325\pi\)
\(48\) −3.39381 + 1.40576i −0.489855 + 0.202904i
\(49\) 0.495478 + 0.495478i 0.0707826 + 0.0707826i
\(50\) −0.301687 −0.0426650
\(51\) 1.46330 − 4.12268i 0.204904 − 0.577291i
\(52\) 10.7158 1.48602
\(53\) 1.96204 + 1.96204i 0.269507 + 0.269507i 0.828902 − 0.559394i \(-0.188966\pi\)
−0.559394 + 0.828902i \(0.688966\pi\)
\(54\) −1.44145 + 0.597069i −0.196157 + 0.0812508i
\(55\) − 2.44557i − 0.329761i
\(56\) 1.13268 + 2.73452i 0.151360 + 0.365416i
\(57\) −6.99022 − 2.89544i −0.925877 − 0.383511i
\(58\) 0.433105 − 1.04561i 0.0568695 − 0.137295i
\(59\) −5.26206 + 5.26206i −0.685062 + 0.685062i −0.961136 − 0.276075i \(-0.910966\pi\)
0.276075 + 0.961136i \(0.410966\pi\)
\(60\) −1.43222 + 1.43222i −0.184898 + 0.184898i
\(61\) −0.346822 + 0.837303i −0.0444061 + 0.107206i −0.944526 − 0.328436i \(-0.893478\pi\)
0.900120 + 0.435642i \(0.143478\pi\)
\(62\) −0.626380 − 0.259455i −0.0795504 − 0.0329508i
\(63\) −1.80017 − 4.34599i −0.226800 − 0.547543i
\(64\) 5.89773i 0.737216i
\(65\) 5.18606 − 2.14814i 0.643252 − 0.266444i
\(66\) 0.553532 + 0.553532i 0.0681351 + 0.0681351i
\(67\) −6.69889 −0.818399 −0.409200 − 0.912445i \(-0.634192\pi\)
−0.409200 + 0.912445i \(0.634192\pi\)
\(68\) −5.84541 − 5.27096i −0.708860 − 0.639198i
\(69\) 0.913326 0.109952
\(70\) 0.535411 + 0.535411i 0.0639938 + 0.0639938i
\(71\) 0.222439 − 0.0921372i 0.0263986 − 0.0109347i −0.369445 − 0.929253i \(-0.620452\pi\)
0.395844 + 0.918318i \(0.370452\pi\)
\(72\) 2.21028i 0.260484i
\(73\) −2.47116 − 5.96591i −0.289228 − 0.698257i 0.710759 − 0.703436i \(-0.248352\pi\)
−0.999987 + 0.00517825i \(0.998352\pi\)
\(74\) 1.42304 + 0.589441i 0.165425 + 0.0685211i
\(75\) −0.406032 + 0.980249i −0.0468846 + 0.113189i
\(76\) −9.62592 + 9.62592i −1.10417 + 1.10417i
\(77\) −4.34022 + 4.34022i −0.494614 + 0.494614i
\(78\) −0.687606 + 1.66003i −0.0778560 + 0.187961i
\(79\) 13.5899 + 5.62912i 1.52898 + 0.633325i 0.979366 − 0.202093i \(-0.0647743\pi\)
0.549615 + 0.835418i \(0.314774\pi\)
\(80\) 1.32492 + 3.19865i 0.148131 + 0.357620i
\(81\) − 0.135560i − 0.0150623i
\(82\) 3.44210 − 1.42577i 0.380117 − 0.157449i
\(83\) −9.82767 − 9.82767i −1.07873 − 1.07873i −0.996624 − 0.0821031i \(-0.973836\pi\)
−0.0821031 − 0.996624i \(-0.526164\pi\)
\(84\) 5.08358 0.554664
\(85\) −3.88561 − 1.37916i −0.421453 − 0.149591i
\(86\) −0.790460 −0.0852375
\(87\) −2.81451 − 2.81451i −0.301748 − 0.301748i
\(88\) 2.66450 − 1.10367i 0.284037 − 0.117652i
\(89\) − 0.395163i − 0.0418872i −0.999781 − 0.0209436i \(-0.993333\pi\)
0.999781 − 0.0209436i \(-0.00666704\pi\)
\(90\) 0.216383 + 0.522395i 0.0228088 + 0.0550653i
\(91\) −13.0162 − 5.39148i −1.36447 − 0.565181i
\(92\) 0.628850 − 1.51818i 0.0655621 − 0.158281i
\(93\) −1.68606 + 1.68606i −0.174836 + 0.174836i
\(94\) −0.490889 + 0.490889i −0.0506314 + 0.0506314i
\(95\) −2.72894 + 6.58825i −0.279983 + 0.675940i
\(96\) −3.33586 − 1.38176i −0.340465 − 0.141025i
\(97\) 3.42855 + 8.27725i 0.348116 + 0.840427i 0.996842 + 0.0794052i \(0.0253021\pi\)
−0.648726 + 0.761022i \(0.724698\pi\)
\(98\) 0.211396i 0.0213542i
\(99\) −4.23471 + 1.75407i −0.425604 + 0.176291i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 85.2.l.a.36.4 yes 24
3.2 odd 2 765.2.be.b.631.3 24
5.2 odd 4 425.2.n.c.274.3 24
5.3 odd 4 425.2.n.f.274.4 24
5.4 even 2 425.2.m.b.376.3 24
17.3 odd 16 1445.2.a.p.1.7 12
17.5 odd 16 1445.2.d.j.866.12 24
17.9 even 8 inner 85.2.l.a.26.4 ✓ 24
17.12 odd 16 1445.2.d.j.866.11 24
17.14 odd 16 1445.2.a.q.1.7 12
51.26 odd 8 765.2.be.b.451.3 24
85.9 even 8 425.2.m.b.26.3 24
85.14 odd 16 7225.2.a.bq.1.6 12
85.43 odd 8 425.2.n.c.349.3 24
85.54 odd 16 7225.2.a.bs.1.6 12
85.77 odd 8 425.2.n.f.349.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.4 ✓ 24 17.9 even 8 inner
85.2.l.a.36.4 yes 24 1.1 even 1 trivial
425.2.m.b.26.3 24 85.9 even 8
425.2.m.b.376.3 24 5.4 even 2
425.2.n.c.274.3 24 5.2 odd 4
425.2.n.c.349.3 24 85.43 odd 8
425.2.n.f.274.4 24 5.3 odd 4
425.2.n.f.349.4 24 85.77 odd 8
765.2.be.b.451.3 24 51.26 odd 8
765.2.be.b.631.3 24 3.2 odd 2
1445.2.a.p.1.7 12 17.3 odd 16
1445.2.a.q.1.7 12 17.14 odd 16
1445.2.d.j.866.11 24 17.12 odd 16
1445.2.d.j.866.12 24 17.5 odd 16
7225.2.a.bq.1.6 12 85.14 odd 16
7225.2.a.bs.1.6 12 85.54 odd 16